Quasi-periodic Dynamics for Multi-dimensional Quasi-linear Schr\"{o}dinger Equations via Resonant Mode Control

TL;DR

This paper establishes the existence of numerous quasi-periodic solutions for multi-dimensional quasi-linear Schrödinger equations by controlling resonant modes using KAM and Fourier analysis techniques.

Contribution

It introduces a novel resonant mode set to manage nonlinear resonances in multi-dimensional quasi-linear Schrödinger equations, combining KAM and Fourier methods.

Findings

Existence of many quasi-periodic solutions proven.

Fourier expansion form of solutions provided.

Frequency shifts estimated in the solutions.

Abstract

This paper focuses on the problem of quasi-periodic solutions for multi-dimensional quasi-linear Schr\"odinger equation. To address the challenge of unbounded perturbations caused by quasi-linear terms in the equation, we define the resonant mode set $\mathcal{K}$ to control nonlinear resonant effects. Combining KAM (Kolmogorov-Arnold-Moser) ( or Nash-Moser ) theory and Fourier analysis methods, we prove that there are plenty of quasi-periodic solutions of the equation. We also present the Fourier expansion form of the solutions and the estimation of frequency shifts.